Implications

This non-implication, Form 186 \( \not \Rightarrow \) Form 359, whose code is 6, is constructed around a proven non-implication as follows:

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 9771, whose string of implications is:
    187 \(\Rightarrow\) 186
  • A proven non-implication whose code is 5. In this case, it's Code 3: 456, Form 187 \( \not \Rightarrow \) Form 100 whose summary information is:
    Hypothesis Statement
    Form 187 <p> Every pair of cardinal numbers has a greatest lower bound (in the usual cardinal ordering.) </p>

    Conclusion Statement
    Form 100 <p> <strong>Weak Partition Principle:</strong>  For all sets \(x\) and \(y\), if \(x\precsim^* y\), then it is not the case that \(y\prec x\). </p>

  • An (optional) implication of code 1 or code 2 is given. In this case, it's Code 2: 1820, whose string of implications is:
    359 \(\Rightarrow\) 20 \(\Rightarrow\) 101 \(\Rightarrow\) 100

The conclusion Form 186 \( \not \Rightarrow \) Form 359 then follows.

Finally, the
List of models where hypothesis is true and the conclusion is false:

Name Statement
\(\cal N12(\aleph_1)\) A variation of Fraenkel's model, \(\cal N1\) Thecardinality of \(A\) is \(\aleph_1\), \(\cal G\) is the group of allpermutations on \(A\), and \(S\) is the set of all countable subsets of \(A\).In \(\cal N12(\aleph_1)\), every Dedekind finite set is finite (9 is true),but the \(2m=m\) principle (3) is false

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